Positional vs Non-Positional Number Systems
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Table of Content:
Positional Value:
Each digit in a number is multiplied by the base raised to the power of its position (from right to left, starting at 0).
Example in Decimal:
543 = 5×10² + 4×10¹ + 3×10⁰ = 500 + 40 + 3 = 543
Example in Binary:
1101 = 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13
| Feature | Positional Number System | Non-Positional Number System |
|---|---|---|
| Definition | A number system where the position of a digit determines its value. | A number system where the value of each symbol is fixed, regardless of position. |
| Value of Digit | Depends on its position (place value) and the base of the system. | Fixed for each symbol; position has no effect on value. |
| Use of Base | Has a defined base (e.g., 2, 10, 16). | No base is used. |
| Examples | Decimal (Base-10), Binary (Base-2), Octal (Base-8), Hexadecimal (Base-16) | Roman Numerals (I, V, X, L, C, D, M), Egyptian numerals, tally marks |
| Symbol Repetition | Symbols can be repeated and reused in different positions. | Often limited repetition; fixed symbols represent values (e.g., X = 10) |
| Ease of Arithmetic Operations | Easy to perform operations like addition, subtraction, etc. | Difficult and less systematic for operations. |
| Compactness | Numbers are represented more compactly. | Representations can be long and redundant. |
| Zero (0) usage | Zero plays a crucial role as a placeholder. | No concept of zero in many non-positional systems. |
| Historical Examples | Hindu-Arabic numeral system, modern computers. | Roman numerals, Babylonian cuneiform. |
| Modern Usage | Used everywhere today in science, math, and computing. | Mostly historical or symbolic use. |
✅ Positional Number System Example (Base-10):
Number: 347
Value:
= 3×10² + 4×10¹ + 7×10⁰
= 300 + 40 + 7
= 347
✅ Non-Positional Number System Example (Roman):
Number: CCCXLVII
= C (100) + C (100) + C (100) + XL (40) + V (5) + I (1) + I (1)
= 100 + 100 + 100 + 40 + 5 + 1 + 1
= 347
? Key Takeaway:
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Positional systems are more powerful, scalable, and practical.
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Non-positional systems are mostly historical and not suitable for complex computation.